Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 9363680, 6 pages
https://doi.org/10.1155/2017/9363680

Research Article

On the Sums of Powers of Chebyshev Polynomials and Their Divisible Properties

Lan Zhang¹ and Wenpeng Zhang²

¹Department of General Education, Xi’an Aeronautical Polytechnic Institute, Xi’an, Shaanxi, China
²School of Mathematics, Northwest University, Xi’an, Shaanxi, China

Correspondence should be addressed to Wenpeng Zhang; wpzhang@nwu.edu.cn

Received 7 August 2017; Accepted 15 November 2017; Published 11 December 2017

Academic Editor: Zhen-Lai Han

Copyright © 2017 Lan Zhang and Wenpeng Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main purpose of this paper is using mathematical induction and the Girard and Waring formula to study a problem involving the sums of powers of the Chebyshev polynomials and prove some divisible properties. We obtained two interesting congruence results involving Fibonacci numbers and Lucas numbers as some applications of our theorem.

1. Discussion and Conclusions

For any integer , the first-kind Chebyshev polynomials and the second-kind Chebyshev polynomials are defined as , , , and , for all . If we write and , then we have

Since Chebyshev polynomials occupy very important position in the theory and application of mathematics, many scholars have studied their various properties and obtained a series of important results. See [1–11]. For example, Li [1] proved some identities involving power sums of and . She obtained some divisibility properties involving Chebyshev polynomials as some applications of these results. Ma and Lv [2] studied the computational problem of the reciprocal sums of Chebyshev polynomials and obtained some identities. Some theoretical results related to Chebyshev polynomials can be found in Ma and Zhang [3], Cesarano [4], Lee and Wong [5], Bhrawy and others (see [6–9]), and Wang and Zhang [10]. Bircan and Pommerenke [11] also obtained many important applications of the Chebyshev polynomials.

In this paper, we will focus on the problem involving the sums of powers of Chebyshev polynomials. These contents not only are widely used in combinatorial mathematics, but also have important theoretical significance for the study of Chebyshev polynomials themselves. Here we will use mathematical induction and the Girard and Waring formula (see [12, 13]) to prove some interesting divisible properties for Chebyshev polynomials. That is, we will prove the following.

Theorem 1. Let and be any positive integers; then we have the congruence

Taking and in and , then we have

where and are Fibonacci numbers and Lucas numbers, respectively.

There are many very interesting and important results related to Fibonacci numbers and Lucas numbers; some of them can be found in Yi and Zhang [14], Ozeki [15], Prodinger [16], Melham [17], and Wang and Zhang [18].

We can also deduce interesting congruence properties involving Fibonacci numbers and Lucas numbers as an application of our theorem. That is, we have the following corollaries.

Corollary 2. For any positive integers and , we have the congruence

Corollary 3. For any positive integers and , we have the congruence

2. Several Simple Lemmas

To complete the proof of our main result, we need several simple lemmas. First we have the following.

Lemma 4. For any positive integer , we have

Proof. If , then and . It is clear that and

So without loss of generality we can assume that . Then from the definition of and binomial theorem we have the identity

From (8), we have the polynomial congruence

or

which implies that

On the other hand, from (8) we also have

which implies that

It is clear that Lemma 4 follows from (11) and (13).

Lemma 5. Let and be any positive integers; then we have the polynomial congruence

Proof. For any fixed positive integer , we prove this polynomial congruence by complete induction for positive integer . Note that , , , , and , so if , then we have

That is to say, Lemma 5 is true for .
If , then note that the identities

we have the polynomial congruence

So Lemma 5 is true for .
Suppose Lemma 5 is true for all integers . That is,

for all .
Then, for , note that the identities

from inductive assumption (18), we have

Now Lemma 5 follows from complete induction.

Lemma 6. For any integers and , we have the congruence

Proof. For integer , note that the identities , ,

It is clear that from (22) we know that, to prove Lemma 6, we need only to prove the polynomial congruence

Now we use complete induction to prove (23). It is clear that polynomial congruence (23) is true for . If , then note that , , and ; we have

So congruence (23) is true for . Suppose (23) is true for all integers . Namely,

for all .
Then, for , note that the identities

from inductive assumption (25) and Lemma 5, we have

This proves Lemma 6 by complete induction.

Lemma 7. For all nonnegative integers and real numbers and , we have the identity

where denotes the greatest integer .

Proof. This formula is given by Girard and Waring, which can be found in [12] or [13].

3. Proof of the Theorem

In this section, we will prove our theorem by mathematical induction. Taking , , and in Lemma 7, note that ; from the definition of and binomial theorem, we have

For any integer , from (29) we have the identity

Taking in (30), we have the identity

From Lemma 4, we know that , so applying Lemma 6 and (31) we can deduce that

That is to say, the theorem is true for .

Suppose that the theorem is true for all integers . That is,

for all integers .

Then, for , from (29) we have

From Lemma 6, we have

Applying inductive assumption (33), we have

Combining (34), (35), (36), and Lemma 6, we can deduce the polynomial congruence

From Lemma 4, we know that , so from (37) we may immediately deduce the polynomial congruence

This completes the proof of our theorem by mathematical induction.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by NSF China (Grant no. 11771351).

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